Truncation bands, unitality, and homomorphisms in truncated Riesz spaces: Boulabiar's problems and beyond
We answer three questions from Boulabiar's survey on truncated Riesz spaces. First, the truncation band projection property coincides with the principal projection property, with no completeness or Archimedean hypothesis needed, and the band projection onto a truncation band is then given explicitly by the truncation itself. Second, a truncation on any Riesz space is unital exactly when its fixed-point set has a supremum. Hence bounded truncations on $KB$-spaces are unital, and a nonunital truncated Banach lattice with bounded truncation contains a closed sublattice copy of $c_0$ and is never an $AL$-space, while on the $AM$ side the smallest truncation unitization norm is always an $M$-norm, though the largest one need not be. Third, a truncation homomorphism whose domain truncation is Archimedean need not be a Riesz homomorphism, as an explicit counterexample built from $c_0$ and $c_0/c_{00}$ shows, but it is one modulo truncation infinitesimals whenever the codomain space is Archimedean.